Why are the Bellman operators contractions?
The inequality
\begin{align}
\left\|T^{\pi} V-T^{\pi} U\right\|_{\infty} &amp; \leq \gamma\|V-U\|_{\infty} \label{1}\tag{1},
\end{align}
where $U$ and $V$ are two value functions, follows from the definition of Bellman policy operator (at slide  16)
\begin{align}
T^{\pi} V(s) 
&amp;\triangleq 
R(s, a)+\gamma \sum_{s^{\prime}} \operatorname{Pr}\left(s^{\prime} \mid s, a\right) V\left(s^{\prime}\right) \\
&amp;=R(s, \pi(s))+\gamma \sum_{s^{\prime}} \operatorname{Pr}\left(s^{\prime} \mid s, \pi(s)\right) V\left(s^{\prime}\right), \; \forall s \in S 
\tag{2}\label{2},
\end{align}
where $\triangleq$ means &quot;defined as&quot;. Note the $\pi$ in the definition, hence the name Bellman policy operator (BPO), and note that the BPO holds for all $s$.
To prove (\ref{1}), first recall that
$$\left\|\mathbf {x} \right\|_{\infty } \triangleq \max _{i}\left|x_{i}\right|.$$
In the case of value functions $V$ and $U$, we have
$$\left\|V - U \right\|_{\infty } \triangleq \max_{s \in S}\left|V(s) - U(s) \right|.$$
Note also that $Pr$ is always non-negative (specifically, between $0$ and $1$).
Successively, we expand the left-hand side of (\ref{1}) by applying the definition (\ref{2}) and using the properties just mentioned
\begin{align}
{\scriptsize
\left\|T^{\pi} V-T^{\pi} U\right\|_{\infty}
= 
\left\|
\left(
R(s, \pi(s))+\gamma \sum_{s^{\prime}} \operatorname{Pr}\left(s^{\prime} \mid s, \pi(s) \right) V\left(s^{\prime}\right) \right) - \left( 
R(s, \pi(s))+\gamma \sum_{s^{\prime}} \operatorname{Pr}\left(s^{\prime} \mid s, \pi(s) \right) U\left(s^{\prime}\right) \right)
\right\|_{\infty} 
}% 
=\\
{\scriptsize
\max_{s \in S}
\left|
\left(
R(s, \pi(s))+\gamma \sum_{s^{\prime}} \operatorname{Pr}\left(s^{\prime} \mid s, \pi(s) \right) V\left(s^{\prime}\right) \right) - \left( 
R(s, \pi(s))+\gamma \sum_{s^{\prime}} \operatorname{Pr}\left(s^{\prime} \mid s, \pi(s) \right) U\left(s^{\prime}\right) \right)
\right|
}% 
= 
\\
\max_{s \in S}
\left|
\gamma \sum_{s^{\prime}} \operatorname{Pr}\left(s^{\prime} \mid s, \pi(s) \right) V\left(s^{\prime}\right) - \gamma \sum_{s^{\prime}} \operatorname{Pr}\left(s^{\prime} \mid s, \pi(s) \right) U\left(s^{\prime}\right)
\right|
= 
\\
\gamma 
\max_{s \in S}
\left|
\sum_{s^{\prime}} \operatorname{Pr}\left(s^{\prime} \mid s, \pi(s) \right) V\left(s^{\prime}\right) - \sum_{s^{\prime}} \operatorname{Pr}\left(s^{\prime} \mid s, \pi(s) \right) U\left(s^{\prime}\right)
\right|
= 
\\
\gamma \max_{s \in S} \left|
\sum_{s^{\prime}} \operatorname{Pr}\left(s^{\prime} \mid s, \pi(s)\right) \left ( V\left(s^{\prime}\right) - U\left(s^{\prime}\right) \right)
\right|
= 
\\
\gamma \max_{s \in S} 
\sum_{s^{\prime}} \operatorname{Pr}\left(s^{\prime} \mid s, \pi(s)\right) \left| V\left(s^{\prime}\right) - U\left(s^{\prime}\right) \right|
\end{align}
Now, note that

$\max _{s^{\prime}}\left|V\left(s^{\prime}\right)-U\left(s^{\prime}\right)\right| = \| V - U \|_{_{\infty}}$
$\alpha x + (1- \alpha) x = x$ for $\alpha \in [0, 1]$
$\left| V\left(s^{\prime}\right) - U\left(s^{\prime}\right) \right| \leq \max_{s' \in S} \left| V\left(s^{\prime}\right) - U\left(s^{\prime}\right) \right|$

NOTE: I am not completely sure about all these steps. I need to think about them a little bit more (but later). I encourage you to try to spot wrong derivations!!